A Third-Order Differential Equation and Starlikeness of a Double Integral Operator

Functions f(z)=z+∑2∞‍anzn that are analytic in the unit disk and satisfy the differential equation f'(z)+αzf''(z)+γz2f'''(z)=g(z) are considered, where g is subordinated to a normalized convex univalent function h. These functions f are given by a double integral operator of the form f(z)=∫01∫01‍G(ztμsν)t-μs-νds dt with G' subordinated to h. The best dominant to all solutions of the differential equation is obtained. Starlikeness properties and various sharp estimates of these solutions are investigated for particular cases of the convex function h.